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Simplifying 3n2 = 6n + 1 Reorder the terms: 3n2 = 1 + 6n Solving 3n2 = 1 + 6n Solving for variable 'n'. Reorder the terms: -1 + -6n + 3n2 = 1 + 6n + -1 + -6n Reorder the terms: -1 + -6n + 3n2 = 1 + -1 + 6n + -6n Combine like terms: 1 + -1 = 0 -1 + -6n + 3n2 = 0 + 6n + -6n -1 + -6n + 3n2 = 6n + -6n Combine like terms: 6n + -6n = 0 -1 + -6n + 3n2 = 0 Begin completing the square. Divide all terms by 3 the coefficient of the squared term: Divide each side by '3'. -0.3333333333 + -2n + n2 = 0 Move the constant term to the right: Add '0.3333333333' to each side of the equation. -0.3333333333 + -2n + 0.3333333333 + n2 = 0 + 0.3333333333 Reorder the terms: -0.3333333333 + 0.3333333333 + -2n + n2 = 0 + 0.3333333333 Combine like terms: -0.3333333333 + 0.3333333333 = 0.0000000000 0.0000000000 + -2n + n2 = 0 + 0.3333333333 -2n + n2 = 0 + 0.3333333333 Combine like terms: 0 + 0.3333333333 = 0.3333333333 -2n + n2 = 0.3333333333 The n term is -2n. Take half its coefficient (-1). Square it (1) and add it to both sides. Add '1' to each side of the equation. -2n + 1 + n2 = 0.3333333333 + 1 Reorder the terms: 1 + -2n + n2 = 0.3333333333 + 1 Combine like terms: 0.3333333333 + 1 = 1.3333333333 1 + -2n + n2 = 1.3333333333 Factor a perfect square on the left side: (n + -1)(n + -1) = 1.3333333333 Calculate the square root of the right side: 1.154700538 Break this problem into two subproblems by setting (n + -1) equal to 1.154700538 and -1.154700538.Subproblem 1
n + -1 = 1.154700538 Simplifying n + -1 = 1.154700538 Reorder the terms: -1 + n = 1.154700538 Solving -1 + n = 1.154700538 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '1' to each side of the equation. -1 + 1 + n = 1.154700538 + 1 Combine like terms: -1 + 1 = 0 0 + n = 1.154700538 + 1 n = 1.154700538 + 1 Combine like terms: 1.154700538 + 1 = 2.154700538 n = 2.154700538 Simplifying n = 2.154700538Subproblem 2
n + -1 = -1.154700538 Simplifying n + -1 = -1.154700538 Reorder the terms: -1 + n = -1.154700538 Solving -1 + n = -1.154700538 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '1' to each side of the equation. -1 + 1 + n = -1.154700538 + 1 Combine like terms: -1 + 1 = 0 0 + n = -1.154700538 + 1 n = -1.154700538 + 1 Combine like terms: -1.154700538 + 1 = -0.154700538 n = -0.154700538 Simplifying n = -0.154700538Solution
The solution to the problem is based on the solutions from the subproblems. n = {2.154700538, -0.154700538}
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